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It's been an incredibly busy week, so I'm just now getting around to writing about Chad's post about how it's not science without graphs. Basically, in a fit of procrastination, he plotted his latest blog traffic stats into a nice little graph, drew a line through the data points, and analyzed the results. It's all very meta of him. But who am I to point fingers? Chad's post made me realize that I am officially an uber-geek. See, back in late January, I got sidelined by the flu and spent a couple of days with a high, spiking fever, unable to do much except moan in between gulps of Theraflu. Bored with flipping channels and the meager offerings of daytime television, I started checking my temperature every hour and recording it, with the aim of plotting it onto a graph when I was done. I had some vague, drug-fogged notion of finding the slope of the tangent curve and thereby practicing my calculus by taking a derivative using a "real-world" example: the rate of change of my body temperature as the fever ran its wicked course.

It didn't quite work out that way: that particular calculus trick only works if the graph gives you a smooth curve. I had so few data points that the result was a series of spiked lines. If I took my temperature every 5 minutes and plotted it out, the end result might have been closer to a curve -- or not. Given the relative crudeness of my digital thermometer, the differences at that point would be so minimal that it probably would have just looked like a straight line. Still, before I started my amateur dabbling into self-taught calculus, I would not have realized that the closer one gets to an infinite number of ever-smaller data points, the more like a curve the resulting graphed data will appear. And it would never have occurred to me to try to create my own real-world calculus problem tracking the rate of change of my own body temperature. Maybe I ended up somewhere other than where I'd intended when I started my little sickbed exercise, but I learned something quite valuable from the experience -- and I'm not likely to forget the "lesson," either.

Real-world examples while learning abstract mathematical principles work for me, despite the recent findings by researchers at Ohio State University that this widespread assumption among educators may be wrong. Ed Yong at Not Exactly Rocket Science has an excellent summation of the study specifics, accompanied by a thought-provoking comment thread. For instance, more than one person said that the so-called "real world" problems one finds in, say, calculus textbooks bear very little resemblance to anything most students would want to solve -- like that silly train analogy that leads off both the New York Times article and Ed's blog post on the study's results. (Jen-Luc Piquant  has her own snide response to when Train A, departing at 6 PM and traveling at 40 MPH toward Station B, will pass Train B, departing at 7 PM and traveling at 50 MPH toward Station A: "When everyone on board is long past caring.") Far from making math "come alive," it's just one more way to make students' eyes glaze over in boredom.

I do not, however, conclude from this that "real world examples don't work." I think it depends on which examples you choose, and how you use them. They are a useful starting point for piquing student interest, but you still have to make the critical connection -- "This relates to that abstract principle, which can be broadly applied to other situations" -- and put in the work to grasp the abstractions.

Jennifer Kaminski, the OSU researcher who spear-headed the study, thinks such an approach obscures the underlying mathematical principle, rather than illuminating it, and actually hinders students' ability to transfer their knowledge to new problems. "They tend to remember the superficial, two trains passing in the night," she told the New York Times. "It's really a problem of our attention getting pulled to superficial information." I can see how that might happen, but I think it's more of a translation problem. Honestly? I sucked at textbook story problems in my K-12 math classes, and received excellent grades in high school geometry and algebra.

But here's the thing: I didn't actually understand the abstractions; I was just blindly following the "rules," manipulating meaningless symbols. And it bored me. I needed some kind of context, just not the equally pointless exercises routinely used in classrooms. The real world examples in textbooks don't really correspond to our daily experiences, or how we might typically approach such a problem. As one of Ed's commenters put it: "If I wanted to know Frankie's and Johnny's ages, I'd ask them, not work out some weird algebra problem." Yet another commenter observed, "'Real-world examples may be treated by students as confusing symbolic concepts that look like real things they know about but act like abstract notions that are defined by the teacher."

I was pleased to read that Kaminski isn't suggesting that we eliminate all real-world examples in classrooms; rather, she thinks that they should augment the abstract principles -- which should be taught first -- rather than being deeply grounded in one specific context. I agree this might increase a student's chances of extrapolating the general principles and applying them to new problems as they arise. Perhaps letting the students choose a real-world problem they'd like to solve -- like my little experiment plotting out my changing rate of body temperature -- is a better way of incorporating a practical context.

You're more likely to pique their interest if they're involved in creating the problems and then figuring out how to solve them -- the "lessons" they learn along the way are more likely to "stick," plus it's a lot more similar to what a working scientist actually does for a living. Is it a calculus problem? A statistical one? How does one go about "translating" that situation into a meaningful mathematical format? This is more of a ground-up approach, akin to taking apart an alarm clock and putting it back together to gain a more comprehensive understanding of how it works. Personally, Jen-Luc would like to see more LOLCats in math and science classes:

This kind of choose-your-own-problems approach also might address the perennial problem of over-generalization -- we all learn differently, and suggesting there is only one correct way to teach a subject like math or physics is likely to leave behind as many students as such a pedagogical approach would advance. And sometimes teachers under-estimate the difficult of new concepts because it's been so long since they learned the material for the first time themselves. As commenter Sam C. said, "Once one has learned something, it's difficult to appreciate what it looks like to someone who hasn't learned it." Something that seems perfectly obvious to the teacher, probably needs to be spelled out, step by step, for many of his/her students.

Case in point: I started my informal calculus "studies" with a DVD lecture series from The Teaching Company. The lectures were pretty good, conceptually: visual elements, real-world examples, but tying them to the abstract principles and then showing how they could be broadly applied. The first thing I learned was how I could (a) use the derivative to figure out the speed of my car from the car's position, and (b) use the integral to figure out how far I'd traveled in my car based on speed. The two are flip sides of the same coin, two different approaches to solving the same problem, depending on the information at one's disposal. And there's a handy real-world context: this is basically what's going on in your car's speedometer and odometer all the time.

Frankly, finding the integral is a labor-intensive process of multiplication and addition take to ridiculous extremes (i.e., infinity). There is a short-cut to the much-harder integral however: if I know both my beginning and ending position, for example, I could just subtract the first from the second to figure out how far I'd traveled. What if I don't know my ending position (and my odometer is broken), just my speed (the velocity function)? Per my DVD instructor, "all" I have to do is figure out which position function generates the known velocity function, and voila! I can do a bit of math-y hocus-pocus to essentially "retrace my steps" backward and use the easier derivative approach. Fair enough, but he never once explained how one goes about finding that position function. There's a lot of them. Still, he insisted it was a simple matter, and silly me -- I believed him.

My DVD instructor lied. It's actually a non-trivial thing for someone just starting out, and/or a bit rusty in their basic algebra and geometry. Don't take my word for it; listen to Johann Bernoulli, a contemporary of Newton and Leibniz who made significant contributions to then-brand-new field of calculus in the 17th century: "But just as much as it is easy to find the differential (derivative) of a given quantity, so it is difficult to find the integral of a given differential," he wrote. "Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not."

Fortunately, I know a lot of physicists and a smattering of mathematicians, most of whom are happy to weigh in now and then with their own insights and "tricks" for the kind of road-block described above. And I'm persistent. I only bring it up because I think it's always interesting to see where different people get hung up when learning new mathematical concepts. Sometimes it's just a language problem, mixing up terminology, or not realizing that you do know what a particular term means -- you just didn't realize that's what your mental concept was called. You hadn't made the connection. Sometimes the instructor has inadvertently left out a step, or doesn't realize that some of his/her students need to be walked through something a bit more carefully.

Because we all learn and think differently -- newsflash: even scientists don't all think and learn alike! -- I'm interested in hearing from readers about similar experiences in their math and science education -- or even their humanities education. I admit, I have an "intuitive" feel for words and writing, and have been guilty in the past of just not understanding why someone couldn't grasp some "trivial" aspect of composition. I've noticed that many "gifted" math sorts can make similar intuitive leaps with numbers. What were your most significant roadblocks? Have you ever stopped to really analyze what happened? How did you overcome them? What are some of the "tricks of the trade" you find useful when applying abstract math principles to "real world" problems?

I think it's a conversation worth having....

Comments

[Pete]

I've often wondered if more people might take to math if it was taught as another language. Maybe I say this because back in the day my verbal scores consistently topped my math scores on standardized tests. But what would happen psychologically if math was framed as the language of nature, complete with it's own dialects, rules of grammar, and alphabets. As for text-books' real-world examples, when I was taking algebra in junior high, I was staring out the door dreaming about designing rockets (early days of the Apollo program), radios, or model airplanes from scratch. I wonder how much more of my attention the teacher would have captured if her examples had to do with subjects like those, even on a simple level, rather than cars passing each other on the freeway. At least they would have touched on the topics of the day. In the end, I got more out of algebra (at least in its simplest forms) as I worked toward my ham-radio license as an adult. I could finally see algebra's practicality as I fiddled with circuit designs!

[Waterdog]

Physics was my favourite subject in high school, and I went on to do my undergraduate degree in it as well. Although I was somebody who grasped most aspects of mathematics relatively easily, and did and do enjoy working on math puzzles, I wasn't terribly interested in pure mathematics as a subject. For my university calculus courses I just did what I had to do to get through them. My goal was to learn only as much mathematics as necessary in order to do physics. Although my interests in science and mathematics are broader now, it was a physics course (electricity and magnetism) that really taught me calculus in a meaningful way. It's a highly practical and useful tool (developed by perhaps the all-time greatest physicist) for understanding the real world.

It was only when I started learning how to conceptualize situations (an annulus of charge of some variable radii, for example) as functional relationships through the language of calculus that I think I truly grasped the meaning of integration at a deep, conceptual level, and saw it's power as a way of describing a real-life situation, and being able to calculate anything about that situation, once you've translated it into mathematical terms. In many ways, this is very much a language, and having made that connection, through practical applications, I now know that I could use calculus to do the same for any physical phenomenon. So, for me, my experience is the opposite of that study you and Chad have both cited.

[Lab Lemming]

The most depressing thing from that academic study article was this:

"Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler did something relatively rare in education research: they performed a randomized, controlled experiment."

Oh noes! Experimentation!

That being said, I think that concrete examples are much more useful in learning how to apply theory than they are in learning the theory.

I've learned and then forgotten calculus about 5 times during the course of my scientific career. So my recollection could be fading. But I seem to recall that the key revelation that I had in geophysics was not learning new and cleverer ways to integrate, it was figuring how to set up the problem so that the resulting equation was something we knew how to deal with.

Of course, this technique requires the ability to be able to perform at least one type of integration. And when it comes to learning how to do the math, I'm not surprised if an algebraic explanation work better than trying to assemble a stack of arbitrarily thin apple slices.

[Heather]

I was dumped in "advanced" math classes beginning in jr. high based on the snap judgment of a well-meaning guidance counselor. My understanding and skills in no way matched those of my peers, and I struggled along thinking I was a dunce at math and that science would never be my thing as a result. In retrospect, I think I would have fared better in the "regular" track. I've seen so many cases in which administrators and instructors assume that an intelligent student will learn regardless of instruction, with unfortunate results. Hey, you're interested enough to seek out a DVD course in calculus, of course that means you'll just kinda "get" how to find that position function.

For a few brief, glorious, months in high school my calculus and physics classes both covered the same content and I actually understood both. Without a concept to anchor the calculus, none of it stuck to my brain. Without a ready ability to work the numbers, the physics all started looking like inexplicable magic. They needed to be together for me to learn either. I remember when the two courses diverged, because that was the point at which I assumed that all of the interesting sciency stuff was beyond me.

I never questioned that assumption until I took an introductory course in acoustics my junior year of college. At which point I promptly signed up for a course in optics and started haunting the physics department. The desire to solve problems that were interesting to me brought the math into focus and I learned. Later, while taking a class on how to teach math to elementary school students, it finally clicked that I wasn't incapable of mathematical thinking. I just needed a chance to explore how it worked before I could practice and master a new concept. There have got to be others in that same position.

I like Pete's idea of math as a language. It makes a lot of sense, and I'd totally take that class. And work on keeping a bilingual household.

[Peter Moore]

Your remarks about the DVD reminded me about my experience at University. At school I'd been keen, and very good, at maths and physics, and pursued astronomy as a hobby. So I ended up going the University of London to study astrophysics. After all, what could be difficult about spending 3 years looking through telescopes, right?

Unfortunately, it was an immense shock. For some reason, at 6th form (UK education tier for 16-17 year olds) I learned about differentiation, but not integration. So I turned up at college and sat in an introductory fluid mechanics course on day one (huh? where's the telescopes?) and saw the professor write some equations on the board. They were gibberish to me. And then he said the immortal words, "and so, obviously, the triple-integral of this becomes...". Obviously. Yeah, right! After that it was just white-noise to me and my promising career as the next Patrick Moore turned instead into year-long party, interrupted by Drama Soc, playing bridge and drinking too much.

[Simplicio]

It seems a pretty common amongst physicists to mention that they learned calculus from their first mechanics course and multi-variate calculus from their first Electromagnetics class rather then from the math classes designed to teach them those topics. The reason, I suspect, is that in early calculus classes its too easy to do well by blindly manipulating the symbols according to the appropriate rules without understanding the underlying abstractions. Physics courses, on the other hand, force you to attach the abstractions to actual physical processes, and require you to have at least some feel for what those symbols actually mean and the motivation for creating them.

[gg]

Pete wrote: "I've often wondered if more people might take to math if it was taught as another language. "

I would highly agree with this. In my experience, a lot of the difficulties students have with the math is making a mental connection between the abstract symbols depicted on the board and the often very intuitive ideas that those symbols represent. Once you understand the 'language', a lot of seemingly difficult mathematical statements become almost embarrassingly trivial. When I teach a math-related course, I make a point of emphasizing this observation to the students.

A couple of examples come to mind: the definition of a 'limit' of a sequence, and very physical vector calculus theorems like Gauss' theorem.

[Oskar]

I'm a real math-head, so I had no trouble at all of understanding both the abstract concepts and the real world applications.

However, I've since worked sporadically as a math-tutor, and I see this problem all the time. Ask a student what the derivative of x^3 with respect to x is and they'll probably get it right, since it's such a simple formula, but ask them what it means and their stumped. In Sweden, where I live, Math and physics are two totally different courses, and when you get to the point in physics where calculus is extremely useful, for some reason the curriculum doesn't assume that you've learned the necessary tools in math. So instead of teaching basic mechanics in terms of calculus, the teacher basically just gives you a formula. As an example, (s)he'll say that if you want to know the position of a stone thrown up in the air with respect to time, use this formula:

p(t) = -g/2*t^2 + v0*t + p0

So the students use it, but really have no concept of what it means. When helping them understand calculus, I usually explain it step by step ("-g is the acceleration due to gravity, right? That's what you get when you differentiate twice. etc.") and it's like a light-bulb goes if inside their head. Suddenly they get a feeling for both calculus and what that damn formula meant.

Because here's the thing: there's an extremely useful, tangible, real-world application for calculus: basic physics! I mean, for Pete's sake, that's what calculus was developed for in the first place! Don't teach it as two subjects, teach them together, as one! You'll learn more, and both will be easier to understand.

[eingram]

When I was learning to solve a loop equation (electronics engineering) of a circuit, a lot of times we would end up with an integral on one side and a differential on the other. Solving such an equation isn't easy (at least for me). Fortunately we were taught Laplace transforms. We could then use plain algebra to solve, and then use the table to transform back to the original domain. The reason I mention this is that there is usually a diferent approach which may be easier (without being wrong). I have often learned concepts that were difficult to grasp by seeing two completely different approaches to the explanation. Hope something in my ramblings makes sense. May the force be against you! (Since it's dark energy).

[Blake Stacey]

Your explanation of your problem puzzles me. I think there's an important distinction to be made here, between the **practical** difficulty of doing a lot of arithmetic and the **conceptual** difficulty of attempting to solve a problem given insufficient information.

Let's say we're given the velocity as a function of time, which we can denote v(t). For the moment, I'm going to assume that this really **is** the velocity, and not just the speed; being a vector quantity, velocity has both magnitude and direction, and speed is the magnitude of the velocity vector. "100 kilometers per hour" is a speed, but "100 kph due north" is a velocity.

Remember doing "absolute values" in algebra class? The absolute value of a number is how far that number is from zero: +1 and -1 both have absolute value 1. Symbolically, this is written with vertical bars:

|-1| = |1| = 1.

Taking the absolute value means forgetting the **direction** you moved away from zero and only remembering the **size** of the step you took. Magnitudes of vectors work the same way:

|100 kph due north| = |100 kph due south| = |100 kph at 35 degrees west of north| = 100 kph.

If you integrate the **speed** as a function of time, you find how far you've traveled, but you won't know where you ended up. To illustrate, imagine I drove for one hour at 100 kph: I've been burning rubber for 100 km, but I might have been making left turns the whole while, so that I've ended where I began. My **velocity vector** has been changing direction, even though it's always remained the same **length**.

OK, now that the terminology is settled, let's do some kinematics.

We know our starting position, x(0), and we're told our velocity function, v(t). If we wanted to figure out how much wear we've put on our car -- the number the odometer measures, which we need to know so we can figure when to change the oil -- we take the magnitude of the velocity function at each point in time to create a new function, the speed:

s(t) = |v(t)|.

Va-va-voom, we integrate the speed.

Suppose we want to figure out where we ended up. We can do this if we know the direction information, i.e., if we have the full velocity function v(t) and not just the speed function s(t). We drove for 1 hour at 100 kph due east, then turned south and drove for 2 hours at 110 kph, etc., etc. Our position at time t is our initial position, x(0), plus the integral from 0 to t of the velocity.

If we **don't** know the direction information, we can't calculate the position function. This is just a fancy way of saying that doing high-speed laps at the Indy 500 will take you to a different place than driving the same speed for the same time in a straight line! It has nothing particular to do with the ease or difficulty of integrating different functions. **That** problem arises because finding integrals, at least the way it's taught in school, is "more art than science" (to coin a phrase). You have to recognize patterns: do we have the derivative of a function divided by the function itself? If so, the answer will involve a logarithm. Can we integrate by parts? Will a clever substitution of variables simplify the problem?

More recently, the mathematicians have figured out how to distill this "art" into something a computer can do, and integrals.wolfram.com (for example) often does a pretty good job. I bet Johann Bernoulli would have gotten a kick out of that!

[Jennifer Ouellette]

I think Blake is getting at a similar point to the one I was trying to make, namely, that the way integrals are traditionally taught in school involves being able to recognize patterns -- at least, if one is going to successfully apply those principles. (And I discovered that, in fact, my DVD lectures DID cover this... it was just buried in a much-later lecture, so far removed from the initial discussion of integrals that I failed to make the connection until I watched the entire lecture series a second time.)

But the broader point I was trying to make is that most textbook examples necessarily must be simplified to a ridiculous extent, e.g., the car driving scenario is usually presented in the case of maintaining a constant speed, and/or a case involving constant rate of acceleration. Then you can use the nifty "cheat sheet" included in most standard calculus textbooks to find the relevant function(s). Where it really becomes art is when you try to apply it to a real-world situation where -- as Blake says -- you are missing key information.

And that, I think, is why the "real world examples" included in textbooks aren't always successful in helping students learn the abstract principles. They're not actually "real."

[Blake Stacey]

Yes, a great many of the "word problems" I remember from my schoolhouse days had clearly been invented to showcase a particular type of equation, rather than to illuminate an aspect of the real world. Feynman's story of evaluating textbooks is quite applicable:

http://www.textbookleague.org/103feyn.htm

I think to actually figure out why mathematics is hard for people to learn, we have to be careful not to confuse different difficulties they might have. If the problem involves calculating the position of a car, for example, and the given information is a big table of velocities measured at different times, then there's (a) the conceptual requirement of knowing how position and velocity are related, and (b) the arithmetic requirement of doing a whole bunch of calculations. If the given information is an algebraic expression for v(t), then the task becomes (c) the evaluation of an integral by symbolic methods, which is a different challenge then either of the other two. Realizing that the speed as a function of time isn't enough information to reconstruct the position falls under (a); if we were given s(t), our skills in department (c) would let us integrate it, but lacking an understanding of part (a), we wouldn't be able to tell how our efforts failed.

[Jennifer Ouellette]

Precisely. How we misunderstand things can be very telling, in that respect. And that's assuming we're being honest about what we don't understand. :) I got As in high school algebra because I was really good at plug-n-chug, and thus my teachers didn't realize how much I was missing in terms of true comprehension. I was much too shy and embarrassed by my failure to tell them, and the end result was far too many years spent suffering from severe math-phobia.

[Blake Stacey]

I had a mild version of the same experience in fifth grade, or thereabouts. I went to a Montessori-like private school for three years, which was a great environment and helped me in all sorts of ways, but I don't think they had actually figured out how to teach math. They used a set of classroom materials called "Mortensen Math", which involves a lot of visual manipulation -- shuffling around blocks of squares drawn on the page, which represent terms in algebraic equations and such. It's a nifty conceit, but it didn't really **connect** with anything else. We didn't really even pick up the idea that "x" stands for an unknown number, and you want to find out what x is, or figure out what possible values it might have. There were even Mortensen booklets on "calculus", and in retrospect I realized that the columns and squares we were drawing and the arithmetic operations we were doing to fill in the worksheet-style blank spaces were actually differentiating and integrating polynomials. You can tell a kid that if you see x^n, you fill this other blank with nx^(n-1), but if you don't explain that this operation means finding the slope of a line, and this other one means finding the area under a curve, then it's just no damn good!

The booklets themselves were probably fine, but all such materials have to be employed by teachers who know what the point of the lesson is. In fifth grade, I didn't really know what I was missing; it took about another five years for me to learn enough math on my own that I could tell my formal classes were severely lacking.

Somewhere on the list of the top half-dozen things I'd like to do in the near future is to write a math textbook. This probably goes to show that I am, indeed, intermittently idealistic and perennially geeky.